By the way, a very similar problem was on my REAL GMAT exam last month.

The problem is as follows.
What is the greatest prime factor of (4^17) - (2^28)?

A) 2
B) 3
C) 5
D) 7
E) 11

SPOILER: D

My short cut is below in the spoiler. Do you think it works for all cases?

SPOILER: Step 1:Create equal -> 4^17 - 2^28 is equivalent to 4^17 - 4^14Step 2: Subtract the exponents; 17-14 =3Step 3: Next, take an easier example to get to a 3 exponent difference still using the number 4 as your base.
Example: 4^4 - 4^1 = 256 - 4 = 252Step 4: Now, working backwards from the answer choices, starting with (E), divide 252 by all answer choices and see what is the highest prime factor to divide into 252.
(E) 252/11 = fraction -> Not a prime factor(D) 252/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)(C) 252/5 = fraction -> Not a prime factor
(B) 252/3 = fraction -> Not a prime factor
(A) 252/2 = 126 -> Is a Prime Factor, but not the highest

Oddly enough this works if you solve it for the base of 2 instead lets consider.

Step 1:Create equal -> 4^17 - 2^28 is equivalent to
(2^17)(2^17) - (2^28) which is also equal to (2^34) - (2^28)Step 2: Subtract the exponents; 34-28 = 6
Step 3: Next, take an easier example to get to a 6 exponent difference still using the number 2 as your base.
Example: 2^7 - 2^1 = 128 - 2 = 126Step 4: Now, working backwards from the answer choices, starting with (E), divide 126 by all answer choices and see what is the highest prime factor to divide into 126.
(E) 126/11 = fraction -> Not a prime factor(D) 126/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)(C) 126/5 = fraction -> Not a prime factor
(B) 126/3 = 42 -> Is a Prime Factor, but not the highest
(A) 126/2 = 126 -> Is a Prime Factor, but not the highest

Seems to work with all examples. Am I crazy, or is this really a shortcut...inquiring minds (me) want to know!!!!!